Solutions - Resonances in a linear array - Linear Array of Spiral Antennas

II. Linear Array of Spiral Antennas

2.2 Resonances in a linear array

2.2.5 Solutions

(a) Distr. of surface current density on the spiral arms. (b) Distr. of charge phase on the spiral arms.

Figure 2.15: Incident right hand circularly polarized wave coming from broadside over asymmetrical square spiral at 1.1526 GHz.

The phase difference at the center of the spiral is, then, about 140^◦ and not 0^◦ as in Fig. 2.7(b).

This result shows the effectiveness of this technique.

Now, let us consider the behavior of the antenna for an incident right-handed circularly polarized wave perpendicular to the plane of the antenna. Fig. 2.15 shows this case. We verify that there is no resonance. As in the case of the symmetrical spiral (cf. Fig. 2.15), the phase difference is almost 180^◦ up to near the end of the arm.

Finally, the simulation results of the asymmetrical square spiral in an infinite linear array with 30^◦ of scan angle are shown in Fig. 2.16. The spacing of this array is 10.6 cm, same as in the array presented in Fig. 2.2, hence, grating lobes are expected to be present at 1.887 GHz for a scan angle of 30^◦. We can see that the reflection coefficient was greatly improved, specially at the resonance frequencies (cf. Tab. 2.2). The total gain and co-pol gain (RHC gain) have dips at these frequencies which means that there is no proper radiation. The XpolR at the steering angle is not good at the frequencies of the dips in the co-pol gain, but they are better at frequencies a bit lower.

Fig. 2.17 shows the distribution of surface current of the array at 1.1526 GHz. It appears a weak resonance in the arms of the spiral, but now the phase difference at the center of the spiral is almost 165^◦, which explains the improvement of reflection coefficient at this frequency.

Instead of remaining constant, this phase difference is reduced from the middle of the arms up to the end, spoiling the radiation, seen as a drop in gain, and reducing the XpolR.

Dissipation of current at the end of spiral arm

Another idea, proposed by Steyskal, to eliminate the resonances, was to use a resistance to absorb the current at the end of the arms (Steyskal et al., 2005). A symmetrical spiral with absorbers can also play the same role. Nakano presented a spiral antenna with a strip absorber below the spiral and at the end of the arms (Nakano et al., 2008).

Fig. 2.18 shows the result of an infinite linear array of such spiral with 30^◦ of scan angle.

The dimensions are listed in Tab. 2.6 where rout is the outer radius of the spiral antenna which

(a) Total Gain. (b) RHC Gain.

Figure 2.16: Infinite linear array of asymmetrical square spirals with element spacing 10.6 cm.

30^◦ of scan angle. Grating lobes should appear at 1.887 GHz (cf. Eq. 1.21).

(a) Distr. of surface current density on the spiral arms. (b) Distr. of charge phase on the spiral arms.

Figure 2.17: Infinite linear array of asymmetrical square spirals at 1.1526 GHz. 30^◦ of scan angle.

(a) Array of spiral antenna with absorber. (b) RHC Gain.

Figure 2.18: Infinite array of self-complementary spiral antenna with absorbers (Nakano et al., 2008). 30^◦ of scan angle. According to Eq. 1.21 grating lobes should appear at 1.82 GHz (cf. Tab. 2.6).

Parameter Value (cm)

rout 5

rin, absorber 4.03 habsorber 1.36

delem 11

Table 2.6: Dimensions of infinite array of self-complementary spiral antenna with strip absorbers below it (Nakano et al., 2008).

strip of 0.97 mm width),habsorber is the height of the absorber which is the same as the distance between the spiral and the plane below the spiral, and delem is the spacing of the array used in the simulation.

Effectively, the array of spiral antennas with strip absorber does not present resonances. It has good circular polarization for frequencies higher than 0.85 GHz and good reflection coefficient

for frequencies higher than 0.9 GHz. The main problem is the gain, which is very low, since at low frequencies an important fraction of the power is dissipated in the strip absorber.

Breaking the periodicity of the array

So far we have discussed the presence of resonances in uniform arrays. For the case of nonuniform arrays, Guinvarc’h and Haupt have proposed an array where the positions of the antennas are selected by the use of Genetic Algorithms for a dual polarized linear array (Guinvarc’h and Haupt, 2010). The optimum distribution is represented by the sequence: ‘‘10100111101101001 010101101110000011000110011100111110001001010101101001000011010’’ where “1” represents a spiral of one polarization (RHCP) and “0” represents a spiral of the opposite polarization (LHCP). In our case we will consider only spirals of one polarization (RHCP), hence, the “0”

positions will represent the absence of the spiral.

(a) Linear array. Green, black and blue corresponds to spirals 27, 28 and 36.

(b) RHC Gain.

Figure 2.19: Linear array of 40 two-arm symmetrical Archimedean spirals, similar to array presented in (Guinvarc’h and Haupt, 2010), but taking just the RHCP subarray. 30^◦ of scan angle. Diameter of spirals is 14 cm and element spacing is 15.65 cm.

Fig. 2.19 presents the simulation results of such mono polarized array. 40 symmetrical spirals, of section 2.2.1 (diameter of 14 cm, pg. 33), was used and the spacing was 15.65 cm. Spiral 36 represents the usual behavior of the spirals of the array, not presenting any peak. Spirals 27 and 28 are the atypical cases. It can be seen that, around 0.79 GHz and 0.9 GHz, there are some peaks in the reflection coefficient of the spirals 27 and 28. These peaks do not necessarily correspond to the Steyskal’s resonance frequencies for these spirals which are 0.84 GHz and 0.95 GHz (see Tab. 2.3). The spirals 27 and 28 correspond to a portion of the array where there are five consecutive spirals, which make them act as a small uniform array inducing the resonances.

Since only these spirals (2 out of 40) present this problem, the gain and XpolR are not affected.

Then, in general, the resonances are not present when nonuniform arrays are used. This effect can be thought of as having different impedances at the end of the arm of the spiral (see transmission line model of Fig. 2.13). This is so because the coupling at the right side of the spiral is different from the left side, hence, the reflected waves, at the end of the arms, do not have the same phase which reduces the resonance. Besides, the non uniformity of the array does not allow the coupling between the spirals to be reinforced and become stronger.

Connecting the spirals

Fig. 2.20 shows a linear array of 14 adjacent and electrically connected square spirals. Each spiral can be inscribed in a circle with a diameter of 14 cm, has 4 turns and its arm length is 84.2 cm. The spacing is 9.6 cm and the scan angle is θ = 30^◦ which means that we can expect grating lobes at 2.09 GHz. We will only consider the spirals at the middle of the array to avoid edge effects.

Two strong peaks at 0.87 GHz and 1.12 GHz appear in the reflection coefficient of the spirals placed right in the middle of the array. Since the arms are connected we can consider that the total arm length is 168.4 cm to calculate the Steyskal’s resonances (cf. Eq. 2.1). Tab. 2.7 presents the Steyskal’s resonances near these two strong peaks showing that they are possibly due to the Steyskal’s resonance. The peaks appear just at the lower part of the bandwidth.

λ/2 multiple m 9 - 10 12 - 13

Observed (GHz) - 0.87 - - 1.12 -

Estimated (GHz) 0.8 - 0.89 1.07 - 1.16

Table 2.7: Resonance frequencies for a linear array of connected square spirals. Total arm length is 168.4 cm.

Additionally, we can see that the XpolR is really poor. The large peaks in the XpolR correspond to frequencies a bit lower than the Steyskal’s resonances when we consider the arm length as being just 84.2 cm (as in a non connected spiral). The XpolR at the Steyskal’s resonance frequencies are well below 15 dB. The dips in the RHC gain corresponds, again, more or less, to the Steyskal’s resonance of the non connected square spiral, up to 1.6 GHz. This reveals that the resonances in this array are not completly gone when we connect the square spirals.

Now, consider Fig. 2.21 where there are 14 Archimedean spirals connected. The diameter of each spiral is 14 cm and the arm length is 92.1 cm, before connecting them. This time, there is just a strong peak in the |S11| at around 0.72 GHz in one of the spirals located at the center of the linear array. This would correspond to one of the Steyskal’s resonances, considering an

(a) Linear array. (b) RHC Gain.

Figure 2.20: Linear array of 14 symmetrical square connected spiral antennas. Diameter of 14 cm and element spacing of 9.6 cm. 30^◦ of scan angle. According to Eq. 1.21 grating lobes should appear at 2.09 GHz. Spirals 7 and 8 are at the middle of the array.

arm length of 184.2 cm (twice, due to the connection) at 0.73 GHz (see Eq. 2.1 for m=9). The XpolR of the array is almost everywhere below 15 dB. On the contrary, the RHC gain is more stable than the connected square spirals case.

The effectivenes of the connection is better in the case of the Archimedean spirals where the connection permits a smoother transition from the spiral arms to free space. As in the case of a horn antenna, the progressive transition from the feed point to open space provides a better matching between the transmission line impedance and the free open space. This, in turn, produces reflected waves back to the source with lower intensity and different phases which destroy the possible resonances (see Fig. 2.22). This does not work well at low frequencies because the aperture is too small compared with the wavelength.

(a) Linear array. (b) RHC Gain.

Figure 2.21: Linear array of 14 symmetrical Archimedean connected spirals. 30^◦ of scan angle.

Diameter of spirals is 14 cm and element spacing is 13.6 cm. Grating lobes should appear at 1.47 GHz (cf. Eq. 1.21). Spirals 7 and 8 are at the middle of the array.

Figure 2.22: Example of a progressive transition to open space. The current is reflected at different parts of the aperture. There is no a strong reflection, since the reflections are not in phase.

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